Stable cohomology of alternating groups

The notion of stable cohomology of groups (and algebraic varieties) was introduced by Fedor Bogomolov in the nineties in the course of his birational anabelian geometry program, and the analysis of rationality properties of linear group quotients $V/G$; in fact the subring of unramified elements in the stable cohomology of $G$ provides obstructions to the stable rationality of $V/G$. Closely related notions have also been studied by Jean-Pierre Serre, and even earlier Grothendieck. In the talk we will discuss some methods for and problems with the effective computation of stable group cohomology, and then focus on the computation of the stable cohomology of the alternating groups $A_n$, which is recent joint work with Bogomolov.